# zbMATH — the first resource for mathematics

Stability of regime-switching diffusion systems with discrete states belonging to a countable set. (English) Zbl 1401.93158

##### MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 93D20 Asymptotic stability in control theory 34D20 Stability of solutions to ordinary differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes
Full Text:
##### References:
 [1] W.J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer, New York, 2012. [2] L. Arnold, W. Kliemann, and E. Oeljeklaus, Lyapunov exponents of linear stochastic systems, in Lyapunov Exponents, Springer, Berlin, Heidelberg, 1986, pp. 85–125. · Zbl 0588.60047 [3] P. Baxendale, Invariant measures for nonlinear stochastic differential equations, in Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math. 1486, Springer, Berlin, 1991, pp. 123–140. · Zbl 0741.60047 [4] M. Benaïm, Stochastic Persistence, preprint, , 2018. [5] M. Benaïm and E. Strickler, Random switching between vector fields having a common zero, Ann. Appl. Probab., to appear. [6] M.F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 2004. · Zbl 1078.60003 [7] A. Hening and D.H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), pp. 1893–1942. · Zbl 1410.60094 [8] R.Z. Khas’minskii, Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Theory Probab. Appl., 12 (1967), pp. 144–147, . [9] R.Z. Khasminskii, C. Zhu, and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), pp. 1037–1051. · Zbl 1119.60065 [10] R. Liptser and A.N. Shiryayev, Theory of Martingales, Math. Appl. (Soviet Ser.) 49, Springer, Dordrecht, 2012. · Zbl 0728.60048 [11] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), pp. 45–67. · Zbl 0962.60043 [12] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. · Zbl 1126.60002 [13] D.H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), pp. 2450–2477, . · Zbl 1391.93199 [14] D.H. Nguyen and G. Yin, Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space, Potential Anal., 48 (2018), pp. 405–435. · Zbl 1426.60079 [15] D.H. Nguyen and G. Yin, Switching diffusion with past dependent switching and countable switching space: Existence and uniqueness of solutions, recurrence, and weak stabilization, in Proceedings of the 56th Annual IEEE Conference on Decision and Control (CDC), 2017, pp. 1858–1863. [16] D.H. Nguyen and G. Yin, Recurrence for linearizable switching diffusion with past dependent switching and countable state space, Math. Control Relat. Fields, to appear. · Zbl 1426.60079 [17] D.H. Nguyen, G. Yin, and C. Zhu, Certain properties related to well posedness of switching diffusions, Stochastic Process. Appl., 127 (2017), pp. 3135–3158. · Zbl 1372.60117 [18] J. Shao and F. Xi, Stability and recurrence of regime-switching diffusion processes, SIAM J. Control Optim., 52 (2014), pp. 3496–3516, . · Zbl 1312.60094 [19] A.V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Transl. Math. Monogr. 78, American Mathematical Society, Providence, RI, 1989. · Zbl 0695.60055 [20] F. Xi and G. Yin, Almost sure stability and instability for switching-jump-diffusion systems with state-dependent switching, J. Math. Anal. Appl., 400 (2013), pp. 460–474. · Zbl 1260.93171 [21] F. Xi and L. Zhao, On the stability of diffusion processes with state-dependent switching, Sci. China Ser. A, 49 (2006), pp. 1258–1274. · Zbl 1107.60321 [22] F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), pp. 1789–1818, . · Zbl 1366.60101 [23] G. Yin and F. Xi, Stability of regime-switching jump diffusions, SIAM J. Control Optim., 48 (2010), pp. 4525–4549, . · Zbl 1210.60089 [24] G. Yin, G. Zhao, and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), pp. 1361–1382, . · Zbl 1272.34076 [25] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. · Zbl 1279.60007 [26] X. Zong, F. Wu, G. Yin, and Z. Jin, Almost sure and $$p$$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), pp. 2595–2622, . · Zbl 1390.34230
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.